Again, this is a legal law issue and depends on your jurisdiction, but his case goes nowhere even if you did sign.

Keep in mind that in many places, your employer is in serious trouble with the Labor Department or equivalent if he does not pay you. Any legal action to obtain the salary in lieu, even if successful in collecting it, would be wiped out by fees, fines from the DoL and his obligation to pay you.

For a written contract to be valid it needs to be signed. If you never signed any contracts you have no employment contract for him to base any legal cases against you on.

What country is this?

If you're in California, most of the conditions in the offer letter are unenforceable anyway. You can quit without notice.